SRM UNIVERSITY
RAMAPURAM PART- VADAPALANI CAMPUS, CHENNAI – 600 026
Department of Mathematics
Sub Title: ADVANCED CALCULUS AND COMPLEX ANALYSIS
Sub Code:15 MA102
Unit -IV - ANALYTIC FUNCTIONS
Part – A
1. Cauchy-Riemann equations are
(a) u x  v y and u y  v x
(b) u x  v y and u y  v x
(c) u x  v x and u y  v y
(d) u x  v y and u y  v x
Ans : (a)
2. If f ( z )  u  iv in polar form is analytic then
(a)
v

(b) r
v

(c)
1 v
r 
(d) 
v

3. If f ( z )  u  iv in polar form is analytic then
(a)
v
r
(b) 
u
is
r
Ans : (c)
u
is

1 v
v
v
(c) 
(d)  r
r r
r
r
Ans : (d)
4. A function u is said to be harmonic if and only if
(a) u xx  u yy  0 (b) u xy  u yx  0 (c) u x  u y  0 (d) u x2  u y2  0
Ans : (a)
5. A function f (z ) is analytic function if
(a) Real part of f (z ) is analytic
(b) Imaginary part of f (z ) is analytic
(c) Both real and imaginary part of f (z ) is analytic (d) none of the above
Ans : (c)
6. If u and v are harmonic functions then f ( z )  u  iv is
(a) Analytic function (b) need not be analytic function
(c) Analytic function only at z  0 (d) none of the above
7. If f ( z )  x  ay  i (bx  cy ) is analytic then a,b,c equals to
Ans : (a)
(c) b  1 and a  c (d) a  b  c  1
Ans : (a)
8. A point at which a function ceases to be analytic is called a
(a) Singular point (b) Non-Singular point (c) Regular point (d) Non-regular point
Ans : (a)
9. The function f ( z ) | z | is a non-constant
(a) c  1 and a  b
(b) a  1 and c  b
(a) analytic function (b) nowhere analytic function (c) non-analytic function (d) entire function
Ans : (b)
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10. A function v is called a conjugate harmonic function for a harmonic function u in  whenever
(a) f  u  iv is analytic (b) u is analytic (c) v is analytic (d) f  u  iv is analytic
Ans : (a)
11. The function f ( x  iy )  x  ax y  bxy  cy is analytic only if
3
2
2
3
(a) a  3i, b  3 and c  i (b) a  3i, b  3 and c  i (c) a  3i, b  3 and c  i
(d) a  3i, b  3 and c  i
Ans : (c)
12. There exist no analytic functions f such that
(a) Re f ( z )  y  2 x (b) Re f ( z )  y 2  2 x (c) Re f ( z )  y 2  x 2 (d) Re f ( z )  y  x
Ans : (b)
ax
13. If e cos y is harmonic, then a is
(a) i
(b) 0
(c) -1
(d) 2
Ans : (a)
14. The harmonic conjugate of 2 x  x  3xy is
3
2
(a) x  3x 2 y  y 3 (b) 2 y  3x 2 y  y 3 (c) y  3x 2 y  y 3 (d) 2 y  3x 2 y  y 3 Ans : (b)
15. The harmonic conjugate of u ( x, y )  2 x(1  y ) is
(a) x 2  y 2  2 x  C (b) x 2  y 2  2 y  C (c) x 2  y 2  2 y  C (d) x 2  y 2  2 y  C
Ans : (d)
16. harmonic conjugate of u ( x, y )  e cos x is
y
(a) e x cos y  C (b) e x sin y  C (c) e y sin x  C (d)  e y sin x  C
Ans : (d)
17. If the real part of an analytic function f (z ) is x 2  y 2  y, then the imaginary part is
(a) 2 xy (b) x 2  2 xy (c) 2 xy  y (d) 2 xy  x
Ans : (d)
18. If the imaginary part of an analytic function f (z ) is 2 xy  y, then the real part is
(a) x 2  y 2  y (b) x 2  y 2  x (c) x 2  y 2  x (d) x 2  y 2  y
Ans : (c)
19. f ( z )  z is differentiable
(b) only at z  0
(a) nowhere
20. f ( z )  z
2
21. f ( z )  z
(d) only at z  1
Ans : (a)
(c) everywhere
(d) only at z  1
Ans : (b)
is differentiable
(b) only at z  0
(a) nowhere
2
(c) everywhere
is
(a) differentiable and analytic everywhere
(b) not differentiable at z  0 but analytic at z  0
(c) differentiable at z  1 and not analytic at z  1 only
(d) differentiable at z  0 but not analytic at z  0
Ans : (d)
 xy
, if z  0;
 2
2
22. If f ( z )   ( x  y )
then f (z ) is
0,
if z  0,

(a) continuous but not differentiable at z  0
(c) analytic everywhere except at z  0
(b) differentiable at z  0
(d) not differentiable at z  0
Ans : (d)
23. f ( z )  e is analytic
z
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(a) only at z  0 (b) only at z  i
(c) nowhere
(d) everywhere
Ans : (d)
24. e (cos y  i sin y) is
x
(a) analytic (b) not analytic
(c) analytic when z  0
(d) analytic when z  i Ans : (b)
(c) analytic when z  0
(d) analytic when z  1 Ans : (a)
25. If f (z ) is analytic, then f (z ) is
(a) analytic (b) not analytic
26. The points at which f ( z ) 
(a) 0 and 1
( z 2  z)
is not analytic are
( z 2  3z  2)
(b) 1 and -1
27. The points at which f ( z ) 
(a) 1 and -1
(c) i and 2
1
is not analytic are
z 1
(b) i and -i
x
x  y2
(b)
2
Ans : (d)
2
(c) 1 and i
28. The harmonic conjugate of u  log
(a)
(d) 1 and 2
y
x  y2
2
(d) -1 and -i
Ans : (b)
x 2  y 2 is
 x
 y
(c) tan 1   (d) tan 1  
x
 y
Ans : (d)
29. If f ( z )  z (2  z ), then f (1  i ) 
(a) 0
(b) i
(c) -i
(d) 2
Ans : (b)
30. If f ( z )  z then f (3  4i ) 
(a) 0
(b) 5
(c) -5
(d) 12
Ans : (b)
31. Critical points of the bilinear transformation w 
(a) a,c (b)
a  bz
are
c  dz
c
c
,  (c)  ,  (d) None of these
d
d
Ans : (c)
32. The points coincide with their transformations are known as
(a) fixed points (b) critical points (c) singular points (d) None of these
a  bz
is a bilinear transformation when
c  dz
(a) ad bc  0 (b) ad  bc  0 (c) ab  cd  0
1
34. w  is known as
z
Ans : (a)
33. w 
(d) None of these
Ans : (b)
(a) inversion (b) translation
(c) rotation
(d) None of these
Ans : (a)
35. w  z   is known as
(a) inversion (b) translation
(c) rotation
(d) None of these
Ans : (b)
36. A translation of the type w  z   where  and  are complex constants, is known as a
(a) translation (b) magnification
(c) linear transformation (d) bilinear transformation
Ans : (c)
37. A mapping that preserves angles between oriented curves both in magnitude and in sense is called a/an .....
mapping.
(a) informal
(b) isogonal
(c) conformal (d) formal
Ans : (c)
38. The mapping defined by an analytic function f (z ) is conformal at all points z except at points where
(a) f ' ( z )  0
(b) f ' ( z )  0
(c) f ' ( z )  0
(d) f ' ( z )  0
Ans : (a)
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39. The fixed points of the transformation w  z 2 are
(a) 0,1 (b) 0,-1 (c) -1,1
(d) –i,i
40. The invariant points of the mapping w 
(a) 1,-1
(b) 0,-1
(c) 0,1
41. The fixed points of w 
(b)  i
(a)  1
(a) confocal ellipses
z
are
2 z
(d) -1,-1
Ans : (c)
z 1
are
z 1
(c) 0,-1
42. The mapping w  z 
Ans : (a)
(d) 0,1
Ans : (b)
1
transforms circles of constant radius into
z
(b) hyperbolas
(c) circles
(d) parabolas
Ans : (a)
1
1
, the image of the line y  in z-plane is
z
4
2
2
2
2
(a) circle u  v  4v  0 (b) circle u  v  4 (c) circle u 2  v 2  2 (d) none of these
43. Under the transformations w 
Ans : (a)
44. The bilinear transformation that maps the points 0, i,  respectively into 0,1,  is w 
(a)
1
z
(b) –z (c) –iz (d) iz
Ans : (c)
45. The bilinear transformation which maps the points z  1, z  0, z  1 of z - plane into w  i, w  0, w  1 of
w  plane respectively is
(a) w  iz
(b) w  z
(c) w  i ( z  1)
(d) none of these
Ans : (a)
Part – B
1. Show that the function f (z) = is no where differentiable.
Solution: Given u+iv = x-iy
u=x
v=-y
ux =1
vx =-1
uy =0
vy =-1
u x vy
C-R equations are not satisfied.
f (z) = is no where differentiable.
2. Show that f (z) =
is differentiable at z=0 but not analytic at z=0.
Solution: Let
=z =
v=0
ux =2x
vx =0
uy =2y
vy = 0
ux = vy and uy = - vx are not satisfied everywhere except at z=0
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So f (z) may be differentiable only at z=0. Now ux,vx,uy,vy are continuous everywhere and in
particular at (0,0).
3. Test the analyticity of the function w=sin z.
Solution: w=f (z) =sin z
u+iv = sin(x+iy)
=sin x cosiy+ cos x siniy
= sin x coshy+i cos x sinhy
u= sin x cushy
v= cos x sinhy
ux = cosx cushy
vx = -sinx sinhy
uy = sinx sinhy
vy = cosx cushy
ux = vy and uy = - vx
C-R equations are satisfied.
The function is analytic.
4. Verify the function 2xy+i(
) is analytic or not .
Solution: u=2xy
v=
ux = 2y
uy = 2x
ux vy and uy
v x = 2x
v y = -2y
- vx
C-R equations are not satisfied.
The function is not analytic.
5. Test the analyticity of the function f (z) = .
Solution: f (z) =
u+iv =
u=
ux =
uy =
=
cosy
cosy
siny
=
(cosy+isiny)
v=
siny
vx =
siny
vy =
cosy
ux = vy and uy = - vx
The function is analytic.
6. If u+iv = is analytic, show that v-iu and –v+iu are also analytic.
Solution: Given u+iv is analytic.
C-R equations are satisfied.
i.e. ux = vy ------------------- (1) and uy = - vx------------------------------(2)
To prove v-iu and –v+iu are also analytic
For this, we have to show that
(i)
ux = vy and -uy = vx
(ii) ux = vy and uy = - vx
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These results follow directly from (1) & (2) by replacing u by v and –v and v –u and u respectively.
v-iu and –v+iu are analytic.
7.Give an example such that u and v are harmonic but u+iv is not analytic.
Solution: Consider the function w= = x-iy
u=x
ux
v=-y
vy ,
The function f(z) is not analytic. But
and
gives u and v are
harmonic.
8.If f (z) = u(x,y) +v(x,y) is an analytic function. Then the curves u(x,y) = c1and v(x,y) =c2 where c1and
c2 are constants are orthogonal to each other.
Solution: If u(x,y) = c1 , then du = 0
But by total differential operator we have
du =
(Say)
Similarly, for the curve v(x,y) =c2 we have
(Say)
For any curve
gives the slope, Now the product of the slopes is
u(x,y) = c1and v(x,y) =c2 intersect at right angles (i.e) they are
orthogonal to
each other.
9.Find the analytic region of f (z) =
Solution: Given f (z) =
u=
v=
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Now ux = vy and uy = - vx
2
=2
-2
x-y=1
= -2
x-y=1
Analytic region of f (z) is x-y=1
10.Find a function w such that w=u+iv is analytic, if u=
.
Solution: Given u=
= 0-i
f (z) = -i
11. Prove that u=
satisfies Laplace’s equation.
Solution: Given u=
u satisfies Laplace’s equation.
12. If u=log (
) find v and f (z) such that f (z) = u+iv is analytic.
Solution: Given u=log (
)
=
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f (z) = 2log z +c
To find the conjugate harmonic v
We know that dv =
=-
[by C – R equations]
dv =
dx
Integrating
V=2
+c
13. Find the critical points for the transformation
Solution: Given
2w
w
Critical points occur at
Also
The critical points occur at
=0
z=
and z =
The critical points occur at z =
14. Find the image of the circle
,
and .
under the transformation w=3z.
Solution: w=3z
u+iv = 3(x+iy)
u=3x
v=3y
x=
y=
Given
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.
maps to a circle in w- plane with centre at the origin and radius 6.
15. Find the fixed points for the following transformation w
Solution: Fixed points are obtained from
f (z) = z
z=
Z=
are the fixed points.
Part – C
1. If f(z) is an analytic function of z, prove that
(i)
=0
(ii)
(iii)
Proof: If z = x+iy then
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=
=
(i).
=
=2
=2
=2
=2
=0
(ii)
=
=
=
=
=2f’ (z)
(iii).
=
=
=
=4
=
2. Prove that the function u =
satisfies laplace’s equation and find the
corresponding analytic function f (z) = u+iv.
Solution: Given u =
+
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+
=
u satisfies Laplace equation.
To find f (z): u is given
Step 1:
+
Step 2:
Step3:
Integrating f (z) =
=
3. Prove that the function v =
is harmonic and determine the corresponding
analytic function of f(z)
Solution: Given v =
Step 1:
+y
Step 2:
Step3:
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Integrating f (z) = -z
To prove v is harmonic
+y
=
4. Prove that the function u =
+1 satisfies laplace’s equation and find the
corresponding analytic function f (z) = u+iv.
Solution: Given u =
+1
= -6x-6
u satisfies Laplace equation.
To find f (z): u is given
Step 1:
Step 2:
Step3:
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Integrating f (z) =
5.
If u=
find the corresponding analytic function f(z) u+iv.
Solution: Given u=
To find f (z): u is given
Step 1:
Step 2:
=
Step3:
Integrating f (z) = tan z
6. Determine the analytic function f(z)=u+iv such that
v=
Solution: f (z) =u+iv ----------------------------- (1)
i f(z) = iu-v
------------------------------(2)
Adding (1) and (2)
F (z) = U+iV
Where F (z) =
Given
,
U=
V=
v=
Step 1:
Step 2:
Step3:
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Integrating F (z) =
(1+i) f (z) =
7. Find the analytic function f(z) = u+iv given that
Solution: 3f (z) = 3u+3iv ---------------------- (1)
2if (2) = 2iu-2v
----------------------- (2)
Adding (1) and (2)
(3+2i) f (z) = (3u-2v) +i (2u+3v)
F (z) = U+iV
Where F (z) = (3+2i) f (z) ,
U=
V=
Given
i.e., V =
Step 1:
Step 2:
Step3:
Integrating F (z) = i cot z
(3+2i) f (z) = i cot z
f (z)
f (z)
8. Find the bilinear transformation that maps the points z = 1, i, -1 into the points w=i, 0, -i
respectively. Hence find the image of
Solution: The bilinear transformations is given by
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w=
w=
is the required bilinear transformation.
To find the image of
Now w=
Since
Put w=u+iv we get
1-2u+
+
1+2u+
The interior of the unit circle
+
(ie)
maps into the half plane a>0 of the w- plane.
9. Find the mobius transformation that maps the points z = 0, 1,
into the points w=-5, -1, 3
respectively. What are the invariant points of the transformation?
Solution: The bilinear transformations is given by
Since
the above relation becomes.
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w+5=3z-5
w=
is the required bilinear transformation.
To get the invariant points, put w=z
z=
Solving for z,
Z
=
=1
The invariant points are z = 1
10. Find the image of
under the transformation.
Solution: Given w = 1/z
z = x+iy and w = u+iv
And
=2
--------------------------- (1)
Substituting x and y values in equation (1), we get
This is the straight line equation in the w-plane.
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11.Show that the transformation w = 1/z transforms circles and straight line in the
circles or straight lines in the w-plane.
Solution: w = 1/z
z-plane into
z = x+iy and w = u+iv
Consider the equation,
If a
----------------------- (1)
equation (1) represents a circle and if a=0, it represents a straight line, substituting the
valus of x and y in (1)
------------------------------------ (2)
If d
0, equation (2) represents a circle and if d=0, it represents a straight line. The various cases
are discussed in detail.
Case (i): When a
d 0
Equation (1) and (2) represents circles in the z-plane and w-plane not passing through the origin.
The transformation w =1/z transforms circles not passing through the origin into circles not
passing through the origin.
Case (ii): When a
d=0
The equation (1) is circle through the origin in z-plane and (2) is a straight line; not passing
through the origin in the w-plane.
Circles passing through the origin in the z-planes maps into the straight lines,
not passing
through the origin in the w-plane.
Case (iii): When a = d 0
Equation (1) represents a straight line not passing through the origin and (2) represents a circle in
the w-plane passing through the origin. Thus lines in the z-plane not passing through the origin
map into circles through the origin in the w-plane.
Case (iv): When a = d= 0
Equation (1) and (2) represents straight lines passing through the origin. Thus the lines through the
origin in the z- plane map into the lines through the origin in the w- plane.
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12. If u=
, v=
prove that u and v are harmonic functions but u+iv is not an
analytic function.
Solution: Given
u=
and
v=
To prove u and v are harmonic
Now
u is harmonic.
Now v=
is harmonic.
Now we show that u+iv is not analytic.
Now
and
It is true from the above relation.
u+iv is not an analytic function.
13. Prove that u =
is harmonic and find its conjugate harmonic.
Solution: Given u =
To prove
Consider u =
Differentiating this w.r.to x and y partially, we get
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u is harmonic.
To find the harmonic conjugate
Let v (x,y) be the conjugate harmonic. Then w = u+iv is analytic.
By C-R equations,
and
=
We have
dv =
dv =
dv =
Integrating, we get
V=
14. . Find the bilinear transformation that maps the points z = -1, 0, 1 into w=0, i, 3i
respectively.
Solution: The bilinear transformations is given by
2w (z-1) = (w-3i) (z+1)
w [2z-2-z-1] = (z+1)(-3i)
w=
is the required bilinear transformation.
15. Find the bilinear transformation that maps the points z = 0, 1,
into the points
w=-1,-2-i, i respectively.
Solution: The bilinear transformations is given by
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Since
the above relation becomes.
2w+2=-zw+iz
W (z+2) = iz-2
w=
is the required bilinear transformation.
Prepared by Mr R.Manimaran,Assistant Professor,Department Of Mathematics,SRM UNIVERSITY,
City Campus,Vadapalani,Chennai-26
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